C₃×#9334 #10627941

order	768
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#9334

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#386
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

398 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3)(4,5,8,12,9,11,18,6)(7,14,16,19,15,10,17,13)(20,21,25,29,26,31,24,30)(22,32,35,42,36,45,34,27)(23,37,39,28,40,43,38,44)(33,41,48,50,49,51,47,46);(4,6,8,11,9,12,18,5)(7,13,17,14,15,19,16,10)(20,22)(21,27,31,42)(23,33)(24,34)(25,35)(26,36)(28,41,44,51)(29,32,30,45)(37,46,43,50)(38,47)(39,48)(40,49);(4,7,9,15)(5,10,11,14)(6,13,12,19)(8,17,18,16)(20,23,26,40)(21,28,31,44)(22,33,36,49)(24,39,25,38)(27,41,42,51)(29,37,30,43)(32,46,45,50)(34,48,35,47)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2322);